Gauge group as a Lie group

Question

Let $\pi\colon P\to M$ be a principal $G$-bundle, where $G$ is a Lie group, the gauge group $\mathscr{G}(P)$ is the group of $G$-automorphism of $P$, that is $G$-equivariant diffeomorphism $\Phi:P\to P$ such that $\pi=\pi\circ\Phi$.

Now I have three questions:

1. I need to know why $\mathscr{G}(P)$ is a Lie group? what is it's smooth structure?
2. Is $\mathscr{G}(P)$ a compact Lie group?
3. I know there is a one to one correspondence between $\mathscr{G}(P)$ and $C^{\infty}(M,Ad P)$, how to show its Lie algebra is $C^{\infty}(M,ad P)$, where $Ad P$ is the associated vector bundle given by conjugate action and $ad P$ is the associated vector bundle given by adjoint action.

Thanks in advance for any help!